Question

The partial differential of normal stress in y-direction in terms of effective stress is given by __________

(a) \(\frac{∂σ_z{‘}}{∂z}\)

(b) \(\frac{∂σ_z{‘}}{∂z}-γ_w\frac{∂h}{∂z}\)

(c) \(\frac{∂σ_z{‘}}{∂z}+γ_w\frac{∂h}{∂z}\)

(d) \(\frac{∂σ_z{‘}}{∂z}*γ_w\frac{∂h}{∂z}\)

The question was asked in a job interview.

This interesting question is from Elasticity topic in portion Elements of Elasticity of Geotechnical Engineering I

Answer 1

The correct choice is (c) \(\frac{∂σ_z{‘}}{∂z}+γ_w\frac{∂h}{∂z}\)

Easy explanation: Since, the normal stress in y-direction in terms of effective stress is given by,

σz= σz’+γw(h-he)

differentiating partially with respect to z,

\(\frac{∂σ_z}{∂z} =   \frac{∂σ_z{‘}}{∂z} +γ_w  \frac{∂(h-h_e)}{∂z}\)

But since \(\frac{∂h_e}{∂z}=0,\)

∴ \(\frac{∂σ_z}{∂z} = \frac{∂σ_z{‘}}{∂z} +γ_w  \frac{∂h}{∂z}.\)